On the asymptotic behavior of sums ∑n≤xf(n){x/n}k

2020 ◽  
Vol 16 (06) ◽  
pp. 1263-1274
Author(s):  
Liuying Wu ◽  
Sanying Shi
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Nondecreasing Function

Let [Formula: see text] be an arbitrary real-valued positive nondecreasing function, in this paper we prove that [Formula: see text] [Formula: see text] where [Formula: see text] is the sum given in the title, [Formula: see text] and [Formula: see text] is a positive integer. This improves the result of Mercier and Nowak.

scholarly journals Sampling parts of random integer partitions: a probabilistic and asymptotic analysis

2015 ◽  
Vol 25 (1) ◽  
pp. 79-95
Author(s):  
Ljuben Mutafchiev
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Asymptotic Analysis ◽  
Integer Partitions ◽  
Random Partition ◽  
Limiting Distributions ◽  
Sampling Plans ◽  
Random Integer ◽  
Sampling Procedures ◽  
The Relationship

Abstract Let λ be a partition of the positive integer n, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of λ at random. They obtained limiting distributions of the multiplicity μn = μn(λ) of the randomly-chosen part as n → ∞. The asymptotic behavior of the part size σn = σn(λ), under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of (μn; σn), as n → ∞, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for (μn; σn). Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of μn and σn.

scholarly journals Selecting the last consecutive record in a record process

2010 ◽  
Vol 42 (03) ◽  
pp. 739-760
Author(s):  
Shoou-Ren Hsiau
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Explicit Expression ◽  
Optimal Stopping ◽  
Continuous Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Maximum Probability ◽  
Optimal Stopping Time ◽  
Bernoulli Random Variables

Suppose that I 1, I 2,… is a sequence of independent Bernoulli random variables with E(I n ) = λ/(λ + n − 1), n = 1, 2,…. If λ is a positive integer k, {I n } n≥1 can be interpreted as a k-record process of a sequence of independent and identically distributed random variables with a common continuous distribution. When I n−1 I n = 1, we say that a consecutive k-record occurs at time n. It is known that the total number of consecutive k-records is Poisson distributed with mean k. In fact, for general is Poisson distributed with mean λ. In this paper, we want to find an optimal stopping time τλ which maximizes the probability of stopping at the last n such that I n−1 I n = 1. We prove that τλ is of threshold type, i.e. there exists a t λ ∈ ℕ such that τλ = min{n | n ≥ t λ, I n−1 I n = 1}. We show that t λ is increasing in λ and derive an explicit expression for t λ. We also compute the maximum probability Q λ of stopping at the last consecutive record and study the asymptotic behavior of Q λ as λ → ∞.

An Extension of Craig's Family of Lattices

2011 ◽  
Vol 54 (4) ◽  
pp. 645-653
Author(s):  
André Luiz Flores ◽  
J. Carmelo Interlando ◽  
Trajano Pires da Nóbrega Neto
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Packing Density ◽  
Sphere Packing ◽  
Construction Technique ◽  
Root Of Unity ◽  
Lattice Construction ◽  
Positive Integers

AbstractLet p be a prime, and let ζp be a primitive p-th root of unity. The lattices in Craig's family are (p –1)-dimensional and are geometrical representations of the integral ℤ[ζp]-ideals 〈1–ζp〉i , where i is a positive integer. This lattice construction technique is a powerful one. Indeed, in dimensions p – 1 where 149 ≤ p ≤ 3001, Craig's lattices are the densest packings known. Motivated by this, we construct (p – 1)(q – 1)-dimensional lattices from the integral ℤ[ζpq]-ideals 〈1 – ζp〉i〈1 – ζq〉j, where p and q are distinct primes and i and j are positive integers. In terms of sphere-packing density, the new lattices and those in Craig's family have the same asymptotic behavior. In conclusion, Craig's family is greatly extended while preserving its sphere-packing properties.

scholarly journals Selecting the last consecutive record in a record process

2010 ◽  
Vol 42 (3) ◽  
pp. 739-760 ◽  
Author(s):  
Shoou-Ren Hsiau
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Explicit Expression ◽  
Optimal Stopping ◽  
Continuous Distribution ◽  
Stopping Time ◽  
Random Variables ◽  
Maximum Probability ◽  
Optimal Stopping Time ◽  
Bernoulli Random Variables

Suppose that I1, I2,… is a sequence of independent Bernoulli random variables with E(In) = λ/(λ + n − 1), n = 1, 2,…. If λ is a positive integer k, {In}n≥1 can be interpreted as a k-record process of a sequence of independent and identically distributed random variables with a common continuous distribution. When In−1In = 1, we say that a consecutive k-record occurs at time n. It is known that the total number of consecutive k-records is Poisson distributed with mean k. In fact, for general is Poisson distributed with mean λ. In this paper, we want to find an optimal stopping time τλ which maximizes the probability of stopping at the last n such that In−1In = 1. We prove that τλ is of threshold type, i.e. there exists a tλ ∈ ℕ such that τλ = min{n | n ≥ tλ, In−1In = 1}. We show that tλ is increasing in λ and derive an explicit expression for tλ. We also compute the maximum probability Qλ of stopping at the last consecutive record and study the asymptotic behavior of Qλ as λ → ∞.

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